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<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist"></a><a class="link" href="inverse_chi_squared_dist.html" title="Inverse Chi Squared Distribution">Inverse
        Chi Squared Distribution</a>
</h4></div></div></div>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">inverse_chi_squared</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
          <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a>   <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">inverse_chi_squared_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
   <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
   <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>

   <span class="identifier">inverse_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> <span class="comment">// Not explicitly scaled, default 1/df.</span>
   <span class="identifier">inverse_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">/</span><span class="identifier">df</span><span class="special">);</span>  <span class="comment">// Scaled.</span>

   <span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// Default 1.</span>
   <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// Optional scale [xi] (variance), default 1/degrees_of_freedom.</span>
<span class="special">};</span>

<span class="special">}}</span> <span class="comment">// namespace boost // namespace math</span>
</pre>
<p>
          The inverse chi squared distribution is a continuous probability distribution
          of the <span class="bold"><strong>reciprocal</strong></span> of a variable distributed
          according to the chi squared distribution.
        </p>
<p>
          The sources below give confusingly different formulae using different symbols
          for the distribution pdf, but they are all the same, or related by a change
          of variable, or choice of scale.
        </p>
<p>
          Two constructors are available to implement both the scaled and (implicitly)
          unscaled versions.
        </p>
<p>
          The main version has an explicit scale parameter which implements the
          <a href="http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution" target="_top">scaled
          inverse chi_squared distribution</a>.
        </p>
<p>
          A second version has an implicit scale = 1/degrees of freedom and gives
          the 1st definition in the <a href="http://en.wikipedia.org/wiki/Inverse-chi-square_distribution" target="_top">Wikipedia
          inverse chi_squared distribution</a>. The 2nd Wikipedia inverse chi_squared
          distribution definition can be implemented by explicitly specifying a scale
          = 1.
        </p>
<p>
          Both definitions are also available in <a href="http://www.wolfram.com/products/mathematica/index.html" target="_top">Wolfram
          Mathematica</a> and in <a href="http://www.r-project.org/" target="_top">The R
          Project for Statistical Computing</a> (geoR) with default scale = 1/degrees
          of freedom.
        </p>
<p>
          See
        </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
              Inverse chi_squared distribution <a href="http://en.wikipedia.org/wiki/Inverse-chi-square_distribution" target="_top">http://en.wikipedia.org/wiki/Inverse-chi-square_distribution</a>
            </li>
<li class="listitem">
              Scaled inverse chi_squared distribution<a href="http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution" target="_top">http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution</a>
            </li>
<li class="listitem">
              R inverse chi_squared distribution functions <a href="http://hosho.ees.hokudai.ac.jp/~kubo/Rdoc/library/geoR/html/InvChisquare.html" target="_top">R
              </a>
            </li>
<li class="listitem">
              Inverse chi_squared distribution functions <a href="http://mathworld.wolfram.com/InverseChi-SquaredDistribution.html" target="_top">Weisstein,
              Eric W. "Inverse Chi-Squared Distribution." From MathWorld--A
              Wolfram Web Resource.</a>
            </li>
<li class="listitem">
              Inverse chi_squared distribution reference <a href="http://reference.wolfram.com/mathematica/ref/InverseChiSquareDistribution.html" target="_top">Weisstein,
              Eric W. "Inverse Chi-Squared Distribution reference." From
              Wolfram Mathematica.</a>
            </li>
</ul></div>
<p>
          The inverse_chi_squared distribution is used in <a href="http://en.wikipedia.org/wiki/Bayesian_statistics" target="_top">Bayesian
          statistics</a>: the scaled inverse chi-square is conjugate prior for
          the normal distribution with known mean, model parameter σ² (variance).
        </p>
<p>
          See <a href="http://en.wikipedia.org/wiki/Conjugate_prior" target="_top">conjugate
          priors including a table of distributions and their priors.</a>
        </p>
<p>
          See also <a class="link" href="inverse_gamma_dist.html" title="Inverse Gamma Distribution">Inverse
          Gamma Distribution</a> and <a class="link" href="chi_squared_dist.html" title="Chi Squared Distribution">Chi
          Squared Distribution</a>.
        </p>
<p>
          The inverse_chi_squared distribution is a special case of a inverse_gamma
          distribution with ν (degrees_of_freedom) shape (α) and scale (β) where
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="serif_italic">α= ν /2 and β = ½</span>
          </p></blockquote></div>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top">
<p>
            This distribution <span class="bold"><strong>does</strong></span> provide the typedef:
          </p>
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">inverse_chi_squared_distribution</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">inverse_chi_squared</span><span class="special">;</span></pre>
<p>
            If you want a <code class="computeroutput"><span class="keyword">double</span></code> precision
            inverse_chi_squared distribution you can use
          </p>
<pre class="programlisting"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">inverse_chi_squared_distribution</span><span class="special">&lt;&gt;</span></pre>
<p>
            or you can write <code class="computeroutput"><span class="identifier">inverse_chi_squared</span>
            <span class="identifier">my_invchisqr</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="number">3</span><span class="special">);</span></code>
          </p>
</td></tr>
</table></div>
<p>
          For degrees of freedom parameter ν, the (<span class="bold"><strong>unscaled</strong></span>)
          inverse chi_squared distribution is defined by the probability density
          function (PDF):
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="serif_italic">f(x;ν) = 2<sup>-ν/2</sup> x<sup>-ν/2-1</sup> e<sup>-1/2x</sup> / Γ(ν/2)</span>
          </p></blockquote></div>
<p>
          and Cumulative Density Function (CDF)
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="serif_italic">F(x;ν) = Γ(ν/2, 1/2x) / Γ(ν/2)</span>
          </p></blockquote></div>
<p>
          For degrees of freedom parameter ν and scale parameter ξ, the <span class="bold"><strong>scaled</strong></span>
          inverse chi_squared distribution is defined by the probability density
          function (PDF):
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="serif_italic">f(x;ν, ξ) = (ξν/2)<sup>ν/2</sup> e<sup>-νξ/2x</sup> x<sup>-1-ν/2</sup> / Γ(ν/2)</span>
          </p></blockquote></div>
<p>
          and Cumulative Density Function (CDF)
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="serif_italic">F(x;ν, ξ) = Γ(ν/2, νξ/2x) / Γ(ν/2)</span>
          </p></blockquote></div>
<p>
          The following graphs illustrate how the PDF and CDF of the inverse chi_squared
          distribution varies for a few values of parameters ν and ξ:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../graphs/inverse_chi_squared_pdf.svg" align="middle"></span>

          </p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../graphs/inverse_chi_squared_cdf.svg" align="middle"></span>

          </p></blockquote></div>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.h0"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.member_functions"></a></span><a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist_ref.dists.inverse_chi_squared_dist.member_functions">Member
          Functions</a>
        </h5>
<pre class="programlisting"><span class="identifier">inverse_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> <span class="comment">// Implicitly scaled 1/df.</span>
<span class="identifier">inverse_chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">);</span> <span class="comment">// Explicitly scaled.</span>
</pre>
<p>
          Constructs an inverse chi_squared distribution with ν degrees of freedom
          <span class="emphasis"><em>df</em></span>, and scale <span class="emphasis"><em>scale</em></span> with default
          value 1/df.
        </p>
<p>
          Requires that the degrees of freedom ν parameter is greater than zero, otherwise
          calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          Returns the degrees_of_freedom ν parameter of this distribution.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          Returns the scale ξ parameter of this distribution.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.h1"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.non_member_accessors"></a></span><a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist_ref.dists.inverse_chi_squared_dist.non_member_accessors">Non-member
          Accessors</a>
        </h5>
<p>
          All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
          functions</a> that are generic to all distributions are supported:
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
        </p>
<p>
          The domain of the random variate is [0,+∞].
        </p>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
            Unlike some definitions, this implementation supports a random variate
            equal to zero as a special case, returning zero for both pdf and cdf.
          </p></td></tr>
</table></div>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.h2"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.accuracy"></a></span><a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist_ref.dists.inverse_chi_squared_dist.accuracy">Accuracy</a>
        </h5>
<p>
          The inverse gamma distribution is implemented in terms of the incomplete
          gamma functions like the <a class="link" href="inverse_gamma_dist.html" title="Inverse Gamma Distribution">Inverse
          Gamma Distribution</a> that use <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>
          and <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a> and their
          inverses <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>
          and <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>:
          refer to the accuracy data for those functions for more information. But
          in general, gamma (and thus inverse gamma) results are often accurate to
          a few epsilon, &gt;14 decimal digits accuracy for 64-bit double. unless
          iteration is involved, as for the estimation of degrees of freedom.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.h3"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.implementation"></a></span><a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist_ref.dists.inverse_chi_squared_dist.implementation">Implementation</a>
        </h5>
<p>
          In the following table ν is the degrees of freedom parameter and ξ is the scale
          parameter of the distribution, <span class="emphasis"><em>x</em></span> is the random variate,
          <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>
          its complement. Parameters α for shape and β for scale are used for the inverse
          gamma function: α = ν/2 and β = ν * ξ/2.
        </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                  <p>
                    Function
                  </p>
                </th>
<th>
                  <p>
                    Implementation Notes
                  </p>
                </th>
</tr></thead>
<tbody>
<tr>
<td>
                  <p>
                    pdf
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: pdf = <a class="link" href="../../sf_gamma/gamma_derivatives.html" title="Derivative of the Incomplete Gamma Function">gamma_p_derivative</a>(α,
                    β/ x, β) / x * x
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: p = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>(α,
                    β / x)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf complement
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: q = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>(α,
                    β / x)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: x = β/ <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>(α,
                    p)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile from the complement
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: x = α/ <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>(α,
                    q)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mode
                  </p>
                </td>
<td>
                  <p>
                    ν * ξ / (ν + 2)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    median
                  </p>
                </td>
<td>
                  <p>
                    no closed form analytic equation is known, but is evaluated as
                    quantile(0.5)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mean
                  </p>
                </td>
<td>
                  <p>
                    νξ / (ν - 2) for ν &gt; 2, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    variance
                  </p>
                </td>
<td>
                  <p>
                    2 ν² ξ² / ((ν -2)² (ν -4)) for ν &gt;4, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    skewness
                  </p>
                </td>
<td>
                  <p>
                    4 √2 √(ν-4) /(ν-6) for ν &gt;6, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis_excess
                  </p>
                </td>
<td>
                  <p>
                    12 * (5ν - 22) / ((ν - 6) * (ν - 8)) for ν &gt;8, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis
                  </p>
                </td>
<td>
                  <p>
                    3 + 12 * (5ν - 22) / ((ν - 6) * (ν-8)) for ν &gt;8, else a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
                  </p>
                </td>
</tr>
</tbody>
</table></div>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.h4"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.references"></a></span><a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist_ref.dists.inverse_chi_squared_dist.references">References</a>
        </h5>
<div class="orderedlist"><ol class="orderedlist" type="1">
<li class="listitem">
              Bayesian Data Analysis, Andrew Gelman, John B. Carlin, Hal S. Stern,
              Donald B. Rubin, ISBN-13: 978-1584883883, Chapman &amp; Hall; 2 edition
              (29 July 2003).
            </li>
<li class="listitem">
              Bayesian Computation with R, Jim Albert, ISBN-13: 978-0387922973, Springer;
              2nd ed. edition (10 Jun 2009)
            </li>
</ol></div>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
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